Convexity counterexample

Here’s a fun counterexample: a function \mathbb{R}^n \to \mathbb{R} that is jointly convex in any n-1 of the variables, but not in all variables at once. The function is

f(x_1,\ldots,x_n) = \frac{1}{2}(n-1.5)\sum_{i=1}^n x_i^2 - \sum_{i < j} x_ix_j

To see why this is, note that the Hessian of f is equal to

\left[ \begin{array}{cccc} n-1.5 & -1 & \cdots & -1 \\ -1 & n-1.5 & \cdots & -1 \\ \vdots & \vdots & \ddots & \vdots \\ -1 & -1 & \cdots & n-1.5 \end{array} \right]

This matrix is equal to (n-0.5)I - J, where I is the identity matrix and J is the all-ones matrix, which is rank 1 and whose single non-zero eigenvalue is n. Therefore, this matrix has n-1 eigenvalues of n-0.5, as well as a single eigenvalue of -0.5, and hence is not positive definite.

On the other hand, any submatrix of size n-1 is of the form (n-0.5)I-J, but where now J is only (n-1) \times (n-1). This matrix now has n-2 eigenvalues of n-0.5, together with a single eigenvalue of 0.5, and hence is positive definite. Therefore, the Hessian is positive definite when restricted to any n-1 variables, and hence f is convex in any n-1 variables, but not in all n variables jointly.


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